Tangents with Parametric Equations

# Tangents with Parametric Equations - Tangents with...

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Tangents with Parametric Equations In this section we want to find the tangent lines to the parametric equations given by, To do this let’s first recall how to find the tangent line to at . Here the tangent line is given by, Now, notice that if we could figure out how to get the derivative from the parametric equations we could simply reuse this formula since we will be able to use the parametric equations to find the x and y coordinates of the point. So, just for a second let’s suppose that we were able to eliminate the parameter from the parametric form and write the parametric equations in the form . Now, plug the parametric equations in for x and y . Yes, it seem silly to eliminate the parameter, then immediately put it back in, but it’s what we need to do in order to get our hands on the derivative. Doing this gives, Now, differentiate with respect to t and notice that we’ll need to use the Chain Rule on the right hand side.

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Let’s do another change in notation. We need to be careful with our derivatives here. Derivatives of the lower case function are with respect to t while derivatives of upper case functions are with respect to x . So, to make sure that we keep this straight let’s rewrite things as follows. At this point we should remind ourselves just what we are after. We needed a formula for or that is in terms of the parametric formulas. Notice however that we can get that from the above equation. Notice as well that this will be a function of t and not x . As an aside, notice that we could also get the following formula with a similar derivation if we needed to, Derivative for Parametric E quations
Why would we want to do this? Well, recall that in the arc length section of the Applications of Integral section we actually needed this derivative on occasion. So, let’s find a tangent line.

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Tangents with Parametric Equations - Tangents with...

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